2021 AN AUTOMATIC ALGORITHM TO DATE THE REFERENCE CYCLE OF THE SPANISH ECONOMY - Documentos de Trabajo
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AN AUTOMATIC ALGORITHM TO DATE
THE REFERENCE CYCLE OF THE
2021
SPANISH ECONOMY
Documentos de Trabajo
N.º 2139
Máximo Camacho, María Dolores Gadea
and Ana Gómez LoscosAN AUTOMATIC ALGORITHM TO DATE THE REFERENCE CYCLE OF THE SPANISH ECONOMY
AN AUTOMATIC ALGORITHM TO DATE THE REFERENCE CYCLE OF THE SPANISH ECONOMY (*) Maximo Camacho (**) UNIVERSITY OF MURCIA María Dolores Gadea (***) UNIVERSITY OF ZARAGOZA Ana Gómez Loscos (****) BANCO DE ESPAÑA (*) We thank seminar participants at Banco de España seminar. M. Camacho and M. D. Gadea are grateful for the support of grants PID2019-107192GB-I00 (AEI/10.13039/501100011033) and 19884/GERM/15, and ECO2017- 83255-C3-1-P and ECO2017-83255-C3-3-P (MICINU, AEI/ERDF, EU), respectively. Data and codes that replicate our results are available from the authors’ websites. The views in this paper are those of the authors and do not represent the views of the Banco de España, the Eurosystem, or the Spanish Business Cycle Dating Committee. (**) Department of Quantitative Analysis, University of Murcia. Campus de Espinardo 30100 Murcia (Spain). Tel: +34 868 887 982 and e-mail: mcamacho@um.es. (***) Department of Applied Economics, University of Zaragoza. Gran Vía, 4, 50005 Zaragoza (Spain). Tel: +34 976 761 842 and e-mail: lgadea@unizar.es. (****) Banco de España, Alcalá, 48, 28014 Madrid (Spain). Tel: +34 91 338 58 17 and e-mail: agomezloscos@bde.es. Documentos de Trabajo. N.º 2139 November 2021
The Working Paper Series seeks to disseminate original research in economics and finance. All papers have been anonymously refereed. By publishing these papers, the Banco de España aims to contribute to economic analysis and, in particular, to knowledge of the Spanish economy and its international environment. The opinions and analyses in the Working Paper Series are the responsibility of the authors and, therefore, do not necessarily coincide with those of the Banco de España or the Eurosystem. The Banco de España disseminates its main reports and most of its publications via the Internet at the following website: http://www.bde.es. Reproduction for educational and non-commercial purposes is permitted provided that the source is acknowledged. © BANCO DE ESPAÑA, Madrid, 2021 ISSN: 1579-8666 (on line)
Abstract This paper provides an accurate chronology of the Spanish reference business cycle by adapting the multiple change-point model proposed by Camacho, Gadea and Gómez Loscos (2021). In that approach, each individual pair of specific peaks and troughs from a set of indicators is viewed as a realization of a mixture of an unspecified number of separate bivariate Gaussian distributions, whose different means are the reference turning points and whose transitions are governed by a restricted Markov chain. In the empirical application, seven recessions in the period from 1970.2 to 2020.2 are identified, which are in high concordance with the timing of the turning point dates established by the Spanish Business Cycle Dating Committee (SBCDC). Keywords: business cycles, turning points, finite mixture models, Spain. JEL classification: E32, C22, E27.
Resumen Este trabajo proporciona una cronología precisa del ciclo económico de referencia de la economía española. Para ello, se adapta el procedimiento de fechado, que considera la posibilidad de múltiples puntos de cambio en la fase del ciclo, propuesto por Camacho, Gadea y Gómez Loscos (2021). En dicho enfoque, cada par individual de picos y valles específicos, obtenidos a partir de un conjunto de indicadores económicos, se considera como una realización de una mixtura de un número no especificado de diferentes distribuciones gaussianas bivariantes, cuyas medias son los puntos de cambio en la fase del ciclo de referencia y cuyas transiciones se rigen por una cadena de Markov restringida. En la aplicación empírica, se identifican siete recesiones en el período comprendido entre 1970.2 y 2020.2. Este fechado es muy similar al establecido por el Comité de Fechado del Ciclo Económico Español. Palabras clave: ciclos económicos, puntos de cambio en la fase del ciclo, modelos de mixturas finitas de distribuciones, España. Códigos JEL: E32, C22, E27.
An automatic algorithm to date the reference cycle of the Spanish economy 2
1 Introduction
Although it seems a truism truth, the business cycle turning points, the dates at which the economy
switches from expansion to recession and vice versa, are not directly observable. Indeed, determin-
ing the reference cycle peaks and troughs is very crucial for policy makers, for businesses and for
the academia because they are used to determine the causes of recessions, to design public policies,
to guide investors and to test competing economic theories, among others.
Aware of this requirement, Martin Feldstein established a Business Cycle Dating Committee
of National Bureau of Economic Research (NBER) scholars and gave it responsibility for business
cycle dating when he took over the institution in 1978. In sum, the committee looks at various
coincident indicators to make informative judgments on when to set the historical dates of the
peaks and troughs of the past US business cycles. Following these guidelines, other countries have
been creating similar committees, such as the Euro Area Business Cycle Dating Committee of the
Center for Economic Policy Research, founded in 2002, and the Spanish Business Cycle Dating
Committee (SBCDC) of the Spanish Economic Association, created in 2014.
Despite the interest in establishing and maintaining a historical chronology of the business
cycle, the dating methodology of the Committees has received some criticism as their decisions
represent the consensus of individuals. Thus, their dating methodology is neither transparent nor
reproducible. In addition, the dating committees date the turning points with a considerable lag,
which last more than two years in some cases. This reduces the interest of the committees’ decisions
to provide real-time assessments of the business cycle changes.
To overcome these drawbacks, this paper codifies in a systematic and transparent way a his-
torical chronology of business cycle turning points for Spain. Following the business cycle concept
of Burns and Mitchell (1946), we consider the reference cycle as a set of dates of wave-like move-
ments occurring at about the same time and in many economic activities. Its computation requires
collecting a number of coincident indicators and determining global change-points, from which the
reference cycle can be extracted using average-then-date or date-then-aggregate methods. The first
case consists on summarizing the coincident information in a single composite indicator from which
the turning points are determined. The second case first identifies turning points in the individual
indicators and looks for common turning points. Although the literature has focused primarily on
average-then-date methods, the date-then-average alternative has recently proved to be successful
in dating the turning points. Examples of the latter are Harding and Pagan (2006, 2016), Chauvet
and Piger (2008), and Stock and Watson (2010, 2014).
Against this background, Camacho et al. (2021) recently proposed a novel date-then-average
methodology that views the reference cycle as a multiple change-point mixture model. They con-
sider that the reference cycle is a collection of increasing change-points (peak-trough dates) that
segment the time span into non-overlapping episodes. With the help of a Markov-switching mix-
ture model representation, the method classifies the dates and the specific turning points, which
are viewed as realization of a mixture of Gaussian distributions.
This method has a number of important advantages over other date-then-average methods.
An automatic algorithm to date the reference cycle of the Spanish economy 3
First, the number of historical change-points are data-driven, so the estimates are not conditioned
by the known occurrence of a phase shift as in Stock and Watson (2010, 2014). Second, the
estimation is simple as it is estimated using standard Bayesian techniques of finite Markov mixture
models. Third, the reference dates are population concepts, which allows us to make inferences in
contrast to Harding and Pagan (2006, 2016). Four, missing data for some coincident indicators is
7 DOCUMENTOas
not a problem
BANCO DE ESPAÑA
they only imply determining some reference cycles from less observations. Five,
DE TRABAJO N.º 2139
the detection of phase changes in real time is a simple as it reduced to a classification problem. This
method was successfully applied to date the US business cycle by using several monthly coincidentestimation is simple as it is estimated using standard Bayesian techniques of finite Markov mixture
models. Third, the reference dates are population concepts, which allows us to make inferences in
contrast to Harding and Pagan (2006, 2016). Four, missing data for some coincident indicators is
not a problem as they only imply determining some reference cycles from less observations. Five,
the detection of phase changes in real time is a simple as it reduced to a classification problem. This
method was successfully applied to date the US business cycle by using several monthly coincident
indicators.
This paper addresses the challenge of applying this methodology to achieve the reference cycle
dates in Spain in a quarterly basis. This poses us several difficulties. Our first challenge was col-
lecting a set of coincident economic indicators with homogeneous quality information throughout
the entire sample given that the Spanish economy has suffered from dramatic transformations in
the last five decades, especially in the seventies and eighties. Our second challenge was adapting
the method to a quarterly basis. In order to date the turning points of each quarterly indicator,
we employ either the so-called BBQ algorithm -proposed by Harding and Pagan (2002), who ex-
tend the monthly algorithm of Bry and Boschan (1971) to a quarterly basis- or the parametric
Markov-Switching procedure -proposed by Hamilton (1989)- depending on the characteristics of
each indicator.
We evaluate the extension of the Camacho et al. (2021) algorithm developed in this paper in
terms of its capacity to generate the SBCDC reference cycle for Spain. The results of this exercise
suggest that our approach is capable of identifying the SBCDC turning points with reasonable
accuracy. Leaving aside the period of the late 1970s and early 1980s, where some deviations occur
with respect to the chronology established by the SBCDC, the method clearly identifies the crisis
of the 1990s, the double dip with which Spain received the impact of the Great Recession and,
finally, the current impact of the Covid-19 pandemic. Thus, we consider that this method can be
very useful to complement and guide the work carried out by the SBCDC in dating the Spanish
business cycles.
The rest of the paper is organized as follows. Section 2 summarizes the methodology proposed
in Camacho et al. (2021) which estimation technique is described in Section 3. Section 4 presents
the empirical application to date the reference cycle of the Spanish reference cycle since the 70s.
Finally, Section 5 concludes.
2 Multiple change-point model
Reference cycle dates can be obtained from the estimates of a multiple change-point model with an
unknown number of K breaks. In practice, we have data in the bivariate time series τ = {τ1 , ..., τN }
of the specific pairs of turning point dates collected from a set of coincident economic indicators.
Each of these turning points, τi , contains a peak date, τiP , and a trough date, τiT , and determines
the start and the end of one specific recessions for a particular coincident indicator. For estimation
purposes, the observed pairs of turning point dates are stacked in ascending order, meaning that
An automatic algorithm to date the reference cycle of the Spanish economy 4
τiP ≤ τi+1
P and τ T ≤ τ T , for i = 1, . . . , N − 1.
i i+1
The approach described in Camacho et al. (2021) assumes that each individual pair of peak and
trough dates in a reference cycle k, τi , is a realization of a density conditioned by τ i−1 = {τ1 , ..., τi−1 }
that depends on the two dimension vector µk = (µPk , µTk )′ and a covariance matrix Σk . The reference
cycle dates are viewed as the means of these distributions, around which the specific turning point
dates dates are clustered. The means and covariances change at unknown time periods breaking the
time span of interest into segments corresponding to the periods occupied by the k = 1, 2, . . . , K
business
8 cycles.
BANCO DE ESPAÑADOCUMENTO DE TRABAJO N.º 2139
We collect the distribution parameters in the reference cycle k in θk = (µk , Σk ) and assume
that these parameters remain constant within each regime and change their values when a regime
change occurs. Then, given that the state is k, τi is drawn from the population given the conditionalthat depends on the two dimension vector µk = (µPk , µTk )′ and a covariance matrix Σk . The reference
cycle dates are viewed as the means of these distributions, around which the specific turning point
dates dates are clustered. The means and covariances change at unknown time periods breaking the
time span of interest into segments corresponding to the periods occupied by the k = 1, 2, . . . , K
business cycles.
We collect the distribution parameters in the reference cycle k in θk = (µk , Σk ) and assume
that these parameters remain constant within each regime and change their values when a regime
change occurs. Then, given that the state is k, τi is drawn from the population given the conditional
density
τi |Ti−1 , θk ∼ p(τi |τ i−1 , θk ) (1)
for k = 1, . . . , K, where p(τi |τ i−1 , θk ) is the Gaussian density N (µk , Σk ). Let us collect all the
unknown distribution parameters in θ = (θ1 , ..., θK ).
This multiple change-point model can be formulated in terms of an integer-valued unobserved
state variable s referred to as the state of the system and that controls the regime changes. In
particular, the discrete random variable is allowed to take values in the set {1, . . . , K} in the whole
sequence of realizations, which are collected in S = (s1 , . . . , sN ). Thus, si = k indicates that τi is
drawn from p(τi |Ti−1 , θk ).
The indicator variable si is modeled as a discrete time, discrete-state first-order Markov chain,
which implies that the probability of a change in regime depends on the past only through the
value of the most recent regime. In this case, the probability of moving from regime l to regime k
at observation i conditional on past regimes and past observations τ i−1 is
Pr(τi = k|si−1 = l, ..., s1 = w, τ i−1 ) = Pr(si = k|si−1 = l) = plk . (2)
In this context, the transition probabilities are constrained to reflect the one-step ahead dy-
namics of the multiple change-point specification. In particular, the order constraints of the break
point model hold when the transition probabilities hold the following restrictions
pll if k = l ̸= M
1 − p if k = l + 1
ll
P r(si = k|si−1 = l) = . (3)
1 if l = K
0 otherwise
The restrictions imply that when the process reaches one regime, for example regime l, it remains in
this regime with probability pll or moves to regime l+1 with a probability 1−pll . The process starts
at regime 1 and moves forward (never backward) to the next regime until it reaches regime K in
which the process stays permanently. Let us collect the transition probabilities {p11 , . . . , pK−1K−1 }
in the vector
Anπ.
automatic algorithm to date the reference cycle of the Spanish economy 5
3 Bayesian estimation
The problem is assigning each specific turning point τi to an unknown reference cycle by making
inference on the unobserved allocation si . To perform the estimation of the parameters collected in
θ and π and the inference on the set of allocations S, Chib (1998) describes a Markov Chain Monte
Carlo (MCMC) algorithm. From the set of observations, the algorithm is implemented using the
Gibbs sampler by simulating the conditional distribution of the parameters given the states, and
the conditional distribution of the states given the parameters.
BANCO DE ESPAÑA 9 DOCUMENTO DE TRABAJO N.º 2139
3.1 Simulation of the parameters
Let us consider sampling the transition probabilities and the parameters of the Gaussian distribu-
tions given the observations and their corresponding allocations, which are assumed to be inde-Gibbs sampler by simulating the conditional distribution of the parameters given the states, and
the conditional distribution of the states given the parameters.
3.1 Simulation of the parameters
Let us consider sampling the transition probabilities and the parameters of the Gaussian distribu-
tions given the observations and their corresponding allocations, which are assumed to be inde-
pendent. Starting with the transition probabilities, Albert and Chib (1993) showed that the full
conditionalAn
distribution
automaticπ|S, τ is independent
algorithm of reference
to date the τ given S.cycle
Following Chib
of the (1998),
Spanish we simulate the5
economy
transition probabilities from π|S using independent Beta priors, pkk ∼ Beta(a, b). The posteriors
remain independent and also follow Beta distributions
3 Bayesian estimation
pkk |Sturning
The problem is assigning each specific ∼ Beta(a + nτkk to
point ,b +
an1), (4)
unknown reference cycle by making
i
inference on the unobserved allocation si . To perform the estimation of the parameters collected in
where nkk is the number of periods the process stays in regime k, with k = 1, . . . , K.1
θ and π and the inference on the set of allocations S, Chib (1998) describes a Markov Chain Monte
To sample θ conditioned to the data and their allocations, we propose an independent normal
Carlo (MCMC) algorithm. From the set of observations, the algorithm is implemented using the
inverse-Wishart prior. Conditional on the mean µk the prior of the inverse covariance matrix is
Gibbs sampler −1 by simulating the conditional distribution of the −1parameters given the states, and
Σ−1
k ∼ W (c 0 , C 0 ) and the posterior is Σ−1
|S, τ, muk ∼ W (c k , C ), where
the conditional distribution of the states kgiven the parameters. k
c k = c 0 + Nk (5)
3.1 Simulation of the parameters ′
Ck = C 0 + (τi − µk ) (τi − µk ) . (6)
Let us consider sampling the transition probabilities
i:s i =k and the parameters of the Gaussian distribu-
tions given the observations and their corresponding allocations, which are assumed to be inde-
In addition, conditional on the covariance matrix, the prior distribution for the means is µk ∼
pendent. Starting with the transition probabilities, Albert and Chib (1993) showed that the full
N (b0 , B0 ), and its posterior is µk |S, τ, Σk ∼ N (bk , Bk ), where
conditional distribution π|S, τ is independent of τ given S. Following Chib (1998), we simulate the
transition probabilities from Bπ|S = � independent
using
−1 −1priors, pkk ∼ Beta(a, b). The posteriors
k B0−1 + Nk (S)ΣBeta
k (7)
remain independent and also follow Beta distributions
bk = Bk (S) B0−1 b0 + Σ−1 k τi . (8)
pkk |S ∼ Beta(a + nkk , b +i:s1),
i =k (4)
Given nthe
where data, one can easily sample means and covariances from their conditional
kk is the number of periods the process stays in regime k, with k = 1, . . . , K.
1 posterior
An automatic algorithm to date the reference cycle of the Spanish economy 6
distributions.
To sample θ conditioned to the data and their allocations, we propose an independent normal
1
Note that NKK+1
inverse-Wishart = 1. Conditional on the mean µk the prior of the inverse covariance matrix is
prior.
3.2
Σ−1 Simulation−1 of the states −1 −1
k ∼ W (c0 , C0 ) and the posterior is Σk |S, τ, muk ∼ W (ck , Ck ), where
Let us consider now the question of sampling the states from the distribution S|θ, π, τ . Using the
c k = c 0 + Nk (5)
properties of the first-order Markov chain, this
distribution can be ′stated as
Ck = C 0 + (τi − µk ) (τi − µk ) . (6)
−1
i:si =k N N
Pr (S|θ, π, τ ) = Pr sN |θ, π, τ Pr si |si+1 , θ, π, τ i , (9)
In addition, conditional on the covariance matrix, the i=1 prior distribution for the means is µk ∼
N (b0 , B0 ), and its posterior is µk |S, τ, Σk ∼ N (bk , Bk ), where
where the last of these distributions is degenerated because we assume that the process starts at
s1 = 1. � −1
Bk = B0−1 + Nk (S)Σ−1 k (7)
Chib (1996) showed that each of the probabilities in expression(9) holds
bk = Bk (S) B0−1 b0 + Σ−1 τi . (8)
P r si |si+1 , θ, π, τ i ∝ Pr (si+1 |sik) Pr si |θ, π, τ i .
i:s =k
(10)
i
The
Givenfirst
theingredient
data, oneincan
thiseasily
expression
samplerefers to and
means the transition
covariancesprobabilities, which are simulated
from their conditional posterior
from the distribution.
distributions.
1 The second ingredient in (10), Pr s |θ, π, τ i , refers to the filtered probabilities and requires
Note that NKK+1 = 1. i
a recursive calculation. Starting from an initial value Pr s0 = k|θ, π, τ 0 , with k = 1, . . . , K, the
10 DOCUMENTO
following
BANCO DE ESPAÑA
two steps are carried out recursively for i = 1, ..., N .2 First, the one-step ahead prediction
DE TRABAJO N.º 2139
with the information up to turning point i − 1
K
The first ingredient in this expression refers to the transition probabilities, which are simulated
The first ingredient in this expression refers to the transition probabilities, which are simulated
The
from first ingredient in this expression refers to the transition probabilities, which are simulated
the distribution.
from the distribution.
fromThe the second
distribution.
An automatic
ingredient algorithm
in (10), to Pr date s |θ,the i , refers to
π, τreference cycle
the of the Spanish
filtered economy
probabilities and requires6
The second ingredient in (10), Pr sii |θ, π, τ ii , refers to the filtered0probabilities and requires
The second
a recursive An ingredient
calculation.
automatic in (10),
Starting
algorithm from Pr date
to si |θ,
an π, τreference
initial
the , refers
value Pr to cycle
s0the= k|θ, filtered
of the
probabilities
π, τSpanish
, with k
economy= and
1, . . . requires
, K, the7
a recursive calculation. Starting from an initial value Pr s0 = k|θ, π, τ 00 , with k = 1, . . . , K, the
afollowing
recursive two calculation.
steps are Starting outfrom an initial for value
i = 1, Pr 2
..., Ns0. =First, k|θ, π,the τ one-step
, with kahead = 1, . prediction
. . , K, the
3.2
following Simulation
two steps are of carried
the states
carried
recursively
out recursively for i = 1, ..., N .22 First, the one-step ahead prediction
following two steps areupcarried
with the information to turning point i − 1for i = 1, ..., N . First, the one-step ahead prediction
out recursively
withusthe
distinct
Let information
cluster
consider separate
now up from
the to turning
question point
otherofclusters
sampling i −of1specific business cycle turning point dates. Notably,
with the information up to turning point i − 1the states from the distribution S|θ, π, τ . Using the
despite
properties theofhuge effort made
the first-order Markovin thischain, area, this the problem
K
distribution of
choosing
can be stated the number of clusters is still
i−1 K
i−1as
unsolved. So, with thePr Praim s = k|θ, π, τ = p Pr s = l|θ, π, τ (11)
siiof= robustness,
k|θ, π, τ i−1 we follow
K lk several
i−1 = l=1 plk Pr si−1 = l|θ, π, τ i−1
i−1different approaches i−1 in the empirical (11)
analysis. Pr si = k|θ, π, τ = l=1 plk N Pr
−1 si−1 = l|θ, π, τ
(11)
Pr (S|θ, π, τ ) = Pr sN |θ, π, l=1 τN Pr si |si+1 , θ, π, τ i , (9)
Among theSecond,
is computed. likelihood-based
when the methods, i-th turning thepointsimplest is i=1 case is the
added, to choose
filtered the model
probability with is the number
updated as
is computed. Second, when the i-th turning point is added, the filtered probability is updated as
is
of computed.
componentsSecond,
follows K thatwhen reaches the thei-th highest
turning marginal point
is added,
likelihood, the
filtered |θ K , MK , from
log p τprobability is updateda set as of
followsthe last of these distributions is degenerated
where Pr si = k|θ, becauseπ, τ i−1 p τ |θ , τ i−1
we assume
i that the process starts at
follows values of {1, ∗ ∗ k
potential Pr. .s.i, = K k|θ, π, τ ii =the
}, where si = k|θ,
Prupper bound π, τ i−1 K pis τspecified
i−1 i−1
i−1 the user, M
i |θk , τ i−1by , is a
K (12)
=
smixture 1. Pr s i = k|θ, π, τ = Pr s i = k|θ,p (τ π, |θ,τ π, τ p τ
) i |θ k , τ , (12)
τ i θ =
i
1 model of K Pr components,
si = k|θ, π, and K is the dK -dimensional p (τi |θ, π, τ i−1 )
vector of its, maximum likelihood (12)
Chib (1996) showed that
K each of the probabilities p (τ in i |θ, π, τ i−1 ) (9) holds
expression
estimated
where p τiparameters.
|θ, π, τ i−1 K Pr s = k|θ, π, τ i−1 p τ |θ , τ i−1 . Now, one can obtain the condi-
|θ, i−1 = K Pr sii = k|θ, π, τ i−1 p τii |θk i−1 . Now, one can obtain the condi-
where Sincep τ i
this π, τ
method =
tends
k=1 to choose models τ i−1
withp a τlarge k , τnumber
i−1 . Now,
of components, we also consider
where p τi |θ, π, τ i−1 = Pk=1 Pr s k|θ,
= π, |θ , τ
tional distribution of the k=1 r si |si+1
states, i
Pr , θ, π,i+1
si |s
i
τ , θ,∝π,Prτ i(s i ,i+1 i k
by|sbackward π, τ i one
i ) Pr si |θ,recursion. . can obtain the condi- (10)
selecting criteria that
tional distribution introduce
of the states, an Pr explicit θ, π, τ i ,term
si |si+1 ,penalty by backwardfor modelrecursion. complexity. The Akaike model
tional The distribution
unconstrained of the MCMCstates,samplerPr si |sdescribedi+1 , θ, π, τabove , by could backward haverecursion.
identifiability problems because
selectionThe procedure is based
unconstrained MCMC on sampler
choosingdescribed the valueabove of K could for which have AIC K = −2 log
identifiability p y|θ K because
problems + 2dK
The
it is The first
subject ingredient
unconstrained
to potentialin this
MCMC expression
label sampler refers
switchingdescribed to
issues. To the
above transition
identifycould the probabilities,
havemodel, identifiability
we use thewhich are
problems simulated
because
identifiability
reaches
it is subject a minimum.to potential Alternatively,
label switching one can choose
issues. Tothe identifymodelthe that minimizes
model, we
use Schwarz’s criterion,
the identifiability
from
it
constraint the distribution.
is subject thatto potential
the drawslabel mustswitching
imply a issues. segmentation To identify of thethe time model, span weinto useK the identifiability
non-overlapping
which is often
constraint thatformulated
the drawsinmust terms implyof minimizing
i BIC = the −2 log p span y|θ K into + dKKlog (N ). 4
episodes,The second
constraint thatµthe
i.e., ingredient
P < draws
µ TWe also decide the number of components by choosing the model with the number of com-
ponents that maximizes the quality of the classification. For this purpose, we define the entropy
as
N
K
ENK = − p (si = k|τi , θ) log p (si = k|τi , θ) , (13)
i=1 k=1
An automatic algorithm to date the reference cycle of the Spanish economy 8
which measures how well the data are classified given a mixture distribution. The entropy takes
the value of 0 for a perfect partition of the data and a positive number that increases as the quality
3.4
of the Data problems
classification deteriorates.
One interesting option is to combine the aim of selecting a model with an optimal number of
The application of finite Markov mixture models to turning point data presents two data challenges.
components (as in the case of the likelihood-based methods) with the aim of obtaining a model
The first is that finite mixtures of Gaussian distributions are defined for continuous data. However,
with a good partition of the data (as proposed by model selection criteria based on entropy mea-
the turning point dates are obtained from coincident indicators that are usually sampled at monthly
sures). For this purpose, we also consider choosing a model whose number of components minimizes
or quarterly frequencies and this generates a discontinuity challenge.
BICK +ENK , which is a metric that penalizes not only model complexity but also misclassification.
For example, when the coincident indicators are sampled at quarterly frequencies, turning
In this context, Kass and Raftery (1995) show that a useful way to compare two models with
points are dates in the format Y Y Y Y.q, which refers to quarter q of year Y Y Y Y . In this case, the
different numbers of clusters Ki and Kj , is to compute twice the natural logarithm of the Bayes
distance between the third month and the forth month of a year is lower than that between the foth
factor Bij , which is the ratio of the two integrated likelihoods that correspond to the models with
quarter of the same year and the first quarter of the following year. To overcome this drawback, we
Ki and Kj clusters, respectively. Fraley and Raftery (2002) point out that this measure can be
propose the transformation of the turning points to dates Y Y Y Y.d, where d = 1/4(q − 1), which
approximated by the difference between the two corresponding BIC
can obviously be changed back to recover the results in the standard quarterly calendar dates.5
The second challenge of this approach is that the individual turning points are collected from
2 log (Bij ) ≈ BIC(Ki ) − BIC(Kj ). (14)
sets of disaggregated economic indicators and some of them might not be available for the full span.
In practice,
This quantitysome indicators
provides are reported
a measure of whetherwiththe
lags
dataexhibiting
increasesmissing observations
or decreases the oddsinofthe latest
a model
months
with Ki while others
clusters are to
relative available
a modelonly
withfor
Kja clusters.
diminished time
Kass andspan because
Raftery they
(1995) are not
propose available
that values
at
of 2the
logbeginning of the2 sample.
(Bij ) less than In our
correspond context,
to weak this is
evidence in rarely a problem
favor of the modelsince
withitKonly impliesvalues
j clusters, that
the probability
between 2 and 6distributions of turning
to positive evidence, points 6atand
between early
10 toand end dates
strong mustand
evidence, be greater
estimated
thanfrom a
10 to
relatively An automatic algorithm to date the reference cycle of the Spanish economy
lower number of observations due to missing data. 8
very strong evidence.
4
In practice, Akaike’s criterion tends to favor more complex models than Schwarz’s criterion since the latter
3.4 Data
penalizes
4 problems
over-fitted models more heavily.
Empirical application
The application of finite Markov mixture models to turning point data presents two data challenges.
In this section we apply the proposed automatic algorithm to generate the reference cycle of the
The first is that finite mixtures of Gaussian distributions are defined for continuous data. However,
Spanish economy. Using the multiple change-point model described above to date the reference
the turning point dates are obtained from coincident indicators that are usually sampled at monthly
cycle turning point dates requires several steps in practice. To serve as an example to other
or quarterly frequencies and this generates a discontinuity challenge.
application, we review these steps in this section.
For example, when the coincident indicators are sampled at quarterly frequencies, turning
points are dates in the format Y Y Y Y.q, which refers to quarter q of year Y Y Y Y . In this case, the
4.1 Collect a set of business cycle indicators
distance between the third month and the forth month of a year is lower than that between the foth
The national
quarter of the Business
same yearCycle Dating
and the Committees
first quarter of theoffollowing
the National Bureau
year. To of Economic
overcome Research
this drawback, we
(NBER) examines
propose the the behavior
transformation of aturning
of the broad set of economic
points to datesindicators.
Y Y Y Y.d, Because
where da=recession
1/4(q −influences
1), which
the
can economy
obviouslybroadly and back
be changed is nottoconfined
recover to
theone sector,
results the standard
in the committees emphasize
quarterly economy-wide
calendar dates.5
measures of economic
The second activity.
challenge of thisTypically,
approachthe committees
is that view real
the individual gross domestic
turning points areproduct (GDP)
collected from
as
setsthe
of single best measure
disaggregated of aggregate
economic indicatorseconomic
and someactivity.
of them might not be available for the full span.
Figure 1some
In practice, showsindicators
that GDP aredoes not increase
reported every
with lags single year
exhibiting in Spain.
missing Instead,
observations there
in the are
latest
particular identifiable
months while episodes
others are during
available onlywhich GDP sharply
for a diminished falls.
time spanThe downturns
because occur
they are notatavailable
around
the periods
at the designated
beginning as recessions
of the sample. In ourbycontext,
the SBCDC,
this is which
rarely aare markedsince
problem as shades
it only areas in that
implies the
6 The committee identified six recessions since the 1970s: the impact of the two oil crises in
graph.
the probability distributions of turning points at early and end dates must be estimated from a
the seventies
relatively andnumber
lower early eighties, the briefdue
of observations crisis
to of the nineties,
missing data. the double dip caused by the global
5
In a similar fashion, when the coincident indicators are sampled at monthly frequency, the dates Y Y Y Y.mm are
transformed into dates Y Y Y Y.d, where d = 1/12(mm − 1).
4 6 Information
Empirical aboutapplication
the Committee and the Spanish turning point dates can be found at
http://asesec.org/CFCweb/en/
BANCO DE ESPAÑA 12 DOCUMENTO DE TRABAJO N.º 2139
In this section we apply the proposed automatic algorithm to generate the reference cycle of the
Spanish economy. Using the multiple change-point model described above to date the reference
cycle turning point dates requires several steps in practice. To serve as an example to otherthe probability distributions of turning points at early and end dates must be estimated from a
relatively lower number of observations due to missing data.
An automatic algorithm to date the reference cycle of the Spanish economy 8
4 Empirical application
An automatic
An automatic algorithm
algorithm to
to date
date the
the reference
reference cycle
cycle of
of the
the Spanish
Spanish economy
economy 98
In
3.4thisData
sectionproblems
we apply the proposed automatic algorithm to generate the reference cycle of the
Spanish economy. Using the multiple change-point model described above to date the reference
3.4
financial Data
The applicationcrisisproblems
inof2008
finiteandMarkov mixture models
the European sovereignto turning
debt crisispointindata 2010 presents two data
and, finally, thechallenges.
economic
cycle turning point dates requires several steps in practice. To serve as an example to other
The firstofisCovid-19
impact that finitepandemic.
mixtures ofTo Gaussian
determine distributions
when a particularare definedrecession
for continuous
begins data. However,
and ends, the
The application
application, of finitethese
we review Markov steps mixture
in thismodels
section.to turning point data presents two data challenges.
the turninguses
committee point dates are obtained from coincident indicatorsdifficult
that aretousually sampled at monthly
The first is that judgments
finite mixtures and of their decisions
Gaussian are, therefore,
distributions are defined for replicate.
continuous data. However,
or quarterly
To overcome frequencies
this and this generates
inconvenient, in this a discontinuity
paper we date challenge.
peaks and troughs from economic indica-
4.1 Collect
the turning pointa dates
set of arebusiness
obtained from cycle indicators
coincident indicators that are usually sampled at monthly
tors For
usingexample,
thefrequencieswhenBBQ
so-called the coincident indicators are sampled at quarterly frequencies, turning
or quarterly and mechanical
this generates algorithm, which
a discontinuity ischallenge.
an implementation of the methodology
The
points
outlined national
areindates Business
Harding in the Cycle
andformat
Pagan Dating
Y(2002).
Y Y Y.q, Committees
which
In short, refersof to
the thequarter
National q ofBureau
year Y YofY Economic
Y Research
. In thispeaks
case, and
the
For example, when the coincident indicators arenonparametric
sampled at quarterly algorithm provides
frequencies, turning
(NBER)
distance of
troughs examines
between
a time thethird
the behavior
bymonth of aand broad
the set
forthof month
economic of aindicators.
year is lower Because a recession
than that between influences
the foth
points are dates in series
the format following
Y Y Y Y.q, three
which steps:
refers(i)toitquarter
estimates q of the
yearpossible
Y Y Y Y .turning
In this points
case, the in
the
the economy
quarter of the
series broadly
same year
by picking and
theand is not
local confined
themaxima
first quarter to one
of sector,
the following the committees
it year. emphasize
Toalternating
overcome this economy-wide
thedrawback, we
distance between the third month and theand forth minima;
month of (ii)a year ensures
is lower than that between troughs and
the foth
measures
propose
the peaks; of
theandeconomic
transformation
(iii) year activity.
it applies Typically,
of atheset turning the committees
pointsoftorules view
datesthat real
Y Y meet gross domestic
Y Y.d,pre-determined
where d = 1/4(q product (GDP)
− 1), of which
quarter of the same and the firstofquarter
censoring of the following year. To overcome this criteria
drawback, the
we
as the
duration single
can obviously and best measure
be changed of
amplitudes of
back aggregate
to recover
phases and economic
the results
complete activity. 7
in the To
cycles. standard
ease of quarterly
comparison, calendar
the dates.and
peaks 5
propose the transformation of the turning points to dates Y Y Y Y.d, where d = 1/4(q − 1), which
trough Figure
Thedates 1 identified
second shows that
challenge withGDP
of this
BBQ does arenot
approach increase
plotted is that every
the
in Figure 1single
individual year
asstandard
red in Spain.
turning
vertical points Instead,
are can
lines. calendar
As there
collected
be seen are
from
5 in
can obviously be changed back to recover the results in the quarterly dates.
particular
sets figure,
the identifiable
of disaggregated episodesin
is noeconomic
there challenge instance during which
indicators and GDPsome of sharply
them is falls.
might The
not be downturns
available occur
for atfull
around
thealgorithm,
span.
The second of this which
approach a SBCDC
is that recession
the individual not turning
identified by the
points areBBQ
collected from
the
which periods
In practice,
picks designated
some
up an as
indicators
additional recessions
are reported
recession byat the
with
the SBCDC, which
lags exhibiting
begging of the are marked as shades areas
missing observations in the latest
eighties. in the
sets of6disaggregated
An automatic economic
algorithm indicators
to date and some
the of themcycle
reference mightofnot thebeSpanish
available for the full span.9
economy
graph.
months The committee
while
Although others isare
GDPindicators identified
the available
most six
only recessions since the
for aindicator,
diminished 1970s:
thetime span thebecause
impact of theythearetwonotoil recession
crises in
available
In practice, some are prominent
reported with lags exhibiting committees
missingemphasize
observations thatina the latest
the
at theseventies
involves beginning
a and of
significantearly
the eighties,
sample.
decline in the
In ourbriefcontext,
economic crisis ofthis
activity the
thatnineties,
is rarely
is athe double
problem
widespread dip caused
since
across it only
the by the global
implies
economy. that
Thus,
months while others are available only for a diminished time span because they are not available
financial crisisfashion,
the5 Inprobability
they aindistributions
2008 and theof European
turning sovereign
points debtand
at sampled
early crisis
endindates2010 and,
must finally,
bedates the
Y Y Yeconomic
estimated fromarea
at theconsider variety of indicators toindicators
determine turning points. In this paper, it we
onlyhave collected
a similar when the coincident are at monthly frequency, the Y.mm
transformed
beginning
into dates
of Ythe
Y Y
sample.
Y.d, where
Ind =our context,
1/12(mm −
this
1).
is rarely a problem since implies that
impact
arelatively
broad ofsetCovid-19
lower
of 51 number pandemic.
specific of To determine
observations
indicators of due
both to when a data.
missing
aggregate particular
activity and recession
employment,beginsand andothers
ends, of the
6
the Information
probability about distributions of turningand
the Committee points the atSpanish
early and end dates
turning point mustdatesbecan estimated
be found from ataa
committee
sectoral uses judgments and theirbothdecisions are,
softtherefore, difficult
TabletoA1 replicate.
relativelynature, which includes harddueand indicators. in the Appendix shows
http://asesec.org/CFCweb/en/
lower number of observations to missing data.
the details of the definitions and the acronyms that will be used in the rest from
4
To overcome this
Empirical application
inconvenient, in this paper we date peaks and troughs of theeconomic
work as indica-
well as
tors using the
the sources andso-called
samples. BBQ mechanical
Overall, algorithm,
the sample begins which is an implementation
in 1970.2 and ends in 2020.3,of thealthough
methodologyeach
4
series
Empirical
outlined
In thishasin Harding
section application
and
we apply
a different lengthPagan
theand (2002).
proposed
combines In short,
automatic the nonparametric
monthlyalgorithm algorithm
to generate
and quarterly provides
the reference
frequencies. peaks
cycle of
The monthly and
the
series
troughs
Spanish
are of a time
economy.into
transformed series
Using by following
the multiple
quarterly by takingthree steps:
change-point
the average (i) it
model estimates
of the described the possible
above to
corresponding turning
date the points
referencein
In this section we apply the proposed automatic algorithm to generate the3 reference
months, although
cycle of theall
the
cycleseries
series have by
turningbeenpicking
point the local
dates
previously maxima
requires
seasonally andsteps
several minima;
to in (ii) it ensures
practice. To alternating
serve the troughs
as an example and
to other
Spanish economy. Using the multiple adjusted
change-point their original
model frequency.
described above to date the reference
the peaks;
application, and
we (iii)
reviewit applies a turning
these steps setinofthis
censoring
section. of rules that
datedmeet pre-determined criteria ofalgo-
the
cycleThe specific
turning business
point datescycle
requires points
several have
steps in been
practice. by applying
To serve as either
an the
example BBQto other
duration
rithm, andthe
when amplitudes
series have of phases andthecomplete cycles.7 Toprocedure
ease of comparison, therepresented
peaks and
application, we review thesea steps
trend,inor this Markov-Switching
section. ,when they are
trough
by dates identified
4.1a composite
Collect aindex
set of with
or BBQrates.
business
growth arecycle
plotted in2Figure
indicators
Figure 1 as red vertical
plots series-specific lines. As
recession can bewhere
episodes, seen ina
the figure, there
series-specific is no instance
recession (inbusiness
red) iniswhich
defineda SBCDC recession
toindicators
be the is not aidentified by totheaBBQ
BBQalgorithm,
4.1 CollectBusiness
The national a set of Cycle Dating cycle
Committees ofperiod from
the National BBQ
Bureau peakof Economic trough
Research or
which
periods picks up an additional recession at the begging of the eighties.
(NBER)with a probability
examines the behaviorof being in recession
of a broad above 0.5.indicators.
set of economic In addition, Figurea recession
Because 3 displaysinfluences
the esti-
The
mated national
Although
probabilityBusiness
GDP is
thatandCycle
the most Dating Committees
prominent
all isindicators indicator,
are intorecession of the
the
in eachNational
committees Bureau
quarter, adding of
emphasize Economic
that
the SBCDC a Research
recession
turning
the economy broadly not confined one sector, the committees emphasize economy-wide
(NBER)
involves
points examines
a significant
as red the behavior
decline
vertical lines. inof a broad
economic set of
activityeconomic
that isindicators.
widespread Because
across a recession
the economy. influences
Thus,
measures of economic activity. Typically, the committees view real gross domestic product (GDP)
the economy
they consider
The SBCDC broadly
a varietyand is not clearly
of indicators confined to one
to determine in sector,
turning the committees
points. In thistheemphasize
paper,
two we oil economy-wide
have
crisescollected
as the single best recessions
measure ofare visible
aggregate economic these
activity.figures. Dating is quite
measures
achallenging
broad set of due
economic
of activity.
51tospecific Typically,
indicators of the
both committees
aggregate view
activity real
and gross domestic
employment, product
and others (GDP)
of a
Figure 1 shows the thatscarce
GDP numberdoes notand low quality
increase every of available
single year in series in the
Spain. seventies
Instead, there when
are
as the was
sectoral
Spain single bestwhich
nature,
under measure
the late of aggregate
includes
years both
of economic
hard
Franco’s and activity.
soft indicators.
dictatorship. The Table
first A1 inofthe
impact the Appendix
oil price shows
rise in
particular identifiable episodes during which GDP sharply falls. The downturns occur at around
the
the Figure
details 1
of shows
the that
definitions GDP and does
the not increase
acronyms that every
will single
be used year
in thein Spain.
rest of Instead,
the work there
as well are
as
the fall
periodsof 1973 only became
designated noticeable
as recessions byinthetheSBCDC,
Spanish whicheconomy areinmarked
1975, and the implementation
as shades areas in the
particular
the
of sources
several identifiable
and samples.
accommodative episodes
Overall, during
the which
sample GDP
begins sharply
in 1970.2falls.and Theendsdownturns
in 2020.3, occur at around
although each
graph. 6 The committee identified six recessions since the 1970s: the impact of the two oil1979,
policies delayed its impact on the real sector until 1977. In crisesthein
the periods
series
economy designated
has suffered
a different from as recessions
length
serious andstructural
combinesby the SBCDC,
monthly
problems, and which
while quarterly
it are hit
was marked
frequencies.
by theassecond
shades areas
The monthly
rise in in the
series
the oil
the seventies and early eighties, the brief crisis of the nineties, the double dip caused by the global
6 The committee
graph.
are transformed
price, plunging intocountry
the identified
quarterly intobyasix recessions
taking the and
recession since
average the
a longof the1970s:
period theweakness
corresponding
of impact3of thelasted
months,
that twoalthough
oiluntil
crises in
all
well
5
In a similar
the seventies
series have fashion,
and
been when the
early eighties,
previously coincident
the brief
seasonally indicators
crisis of
adjusted are sampled
tothe at
nineties,
their monthly
originalthe frequency, the dates Y Y Y
double dip caused by the global
frequency. Y.mm are
into the next decade.
transformed into dates Y Y Y Y.d, where d = 1/12(mm − 1).
6 The specific business cycle turning points have been dated by applying either the BBQ algo-
5
Information
In a similar
The SBCDC about
fashion, the the
when
identifies Committee
twocoincident and1974.4,
thearesignaling
peaks: indicators
in Spanish
sampled atturning
amonthly point
recession dates
frequency,
between can
the1975.1 beand
dates Y Y Y found
Y.mm at
are
1975.2,
http://asesec.org/CFCweb/en/
transformed
rithm, when into
thedates Y Yhave
series Y Y.d,awhere
trend,d=or1/12(mm
the − 1).
Markov-Switching procedure ,when they are represented
7
6
For example, aabout
Information complete
thecycle must last four
Committee andquarters
the at least. Inturning
Spanish addition, point
the minimum
dates duration
can beof afound
recession
at
by
is a composite
two quarters. index or growth rates. Figure 2 plots series-specific recession episodes, where a
http://asesec.org/CFCweb/en/
series-specific recession (in red) is defined to be the period from a BBQ peak to a BBQ trough or
periods13with
BANCO DE ESPAÑA DOCUMENTO DE TRABAJO N.º 2139
a probability of being in recession above 0.5. In addition, Figure 3 displays the esti-
mated probability that all indicators are in recession in each quarter, adding the SBCDC turning
points as red vertical lines.the sources and samples. Overall, the sample begins in 1970.2 and ends in 2020.3, although each
series has a different length and combines monthly and quarterly frequencies. The monthly series
are transformed into quarterly by taking the average of the corresponding 3 months, although all
series have been previously seasonally adjusted to their original frequency.
The specific business cycle turning points have been dated by applying either the BBQ algo-
rithm, when the series have a trend, or the Markov-Switching procedure ,when they are represented
by a composite index or growth rates. Figure 2 plots series-specific recession episodes, where a
series-specific recession (in red) is defined to be the period from a BBQ peak to a BBQ trough or
periods with a probability of being in recession above 0.5. In addition, Figure 3 displays the esti-
mated probability that all indicators are in recession in each quarter, adding the SBCDC turning
points as red vertical lines.
The SBCDC recessions are clearly visible in these figures. Dating the two oil crises is quite
challenging due to the scarce number and low quality of available series in the seventies when
Spain was under the late years of Franco’s dictatorship. The first impact of the oil price rise in
the fall of 1973 only became noticeable in the Spanish economy in 1975, and the implementation
of several accommodative policies delayed its impact on the real sector until 1977. In 1979, the
economy suffered from serious structural problems, while it was hit by the second rise in the oil
price, plunging the country into a recession and a long period of weakness that lasted until well
An automatic algorithm to date the reference cycle of the Spanish economy 10
into the next decade.
The SBCDC identifies two peaks: in 1974.4, signaling a recession between 1975.1 and 1975.2,
and7 For
in example,
1978.3, aresulting in a must
complete cycle recession between
last four quarters 1978.4
at least. and 1979.2.theThe
In addition, application
minimum durationof
of the BBQ
a recession
is two quarters.
algorithm, delays the dating of each of these recessions by a quarter. The algorithm also identifies
a third recession between 1981.2-1981.3, which captures the slow recovery of the Spanish economy
in the face of the second oil price hike. This behavior differs from most of the rest of the European
countries, which underwent a recovery at the beginning of the eighties.
The crisis of the nineties took place between 1992.2 and 1993.2 according to the SBCDC and
is delayed by a quarter if we apply the BBQ algorithm to GDP. It was a short intense recession
after years of sound economic growth. The recession was synchronized with the cyclical phase of
most European countries. A high percentage of series point to a recession: 71% of the 51 series
detect the crisis in the quarters spanning from 1992.3 to 1993.1 and 54% lead or lag its effect.
The global financial crisis and the European debt crisis had a strong impact on the Spanish
economy, which involved two waves of recession separated by a brief interval of recovery. Most
of the series accurately pinpoint the first hit of the crisis (Figures 2 and 3). Specifically, between
2008.3 and 2009.1 (official SBCDC dates) more than two thirds of the series identified the crisis,
although 69% had already detected it in 2008.2. The second impact, dated between 2011.1 and
2013.2 according to SBCDC, is confirmed by 70% of the series analyzed. A non-negligible set of
series, around 50%, did not detect the intermediate recovery period between both impacts.
At the beginning of 2020, the world was suddenly hit by a global pandemic due to the coron-
avirus disease. The pandemic triggered the most severe global economic crisis since World War II in
terms of speed and intensity. To mitigate the propagation of the virus, governments implemented
unprecedented measures, such as restrictions on movement and travel, population lockdowns, and
social distancing in some targeted activities. Official institutions place the recession peak in the
last quarter of 2019 or in February of 2020 at the monthly frequency. The SBCDC places the peak
in the fourth quarter of 2019 and, therefore, the beginning of the crisis in the first quarter of 2020.
More than 70% of the series detect the crisis, although there is a significant percentage of series,
around 50%, that already pointed to a slowdown in the economy that was obviously not related to
Covid-19 pandemic which, being an exogenous shock to the economy, was impossible to predict.
Summing up, around the dates that the SBCDC identifies recessions there is a wide set of
specific14indicators
BANCO DE ESPAÑA
that show a recession. However, there are indicators that either advance (usually
DOCUMENTO DE TRABAJO N.º 2139
related to the industrial sector), delay or extend the duration of recessions (for example, those
associated with the labor market). Furthermore, not all the specific indicators have the same
behavior in each recession.More than 70% of the series detect the crisis, although there is a significant percentage of series,
around 50%, that already pointed to a slowdown in the economy that was obviously not related to
Covid-19 pandemic which, being an exogenous shock to the economy, was impossible to predict.
Summing up, around the dates that the SBCDC identifies recessions there is a wide set of
specific indicators that show a recession. However, there are indicators that either advance (usually
related to the industrial sector), delay or extend the duration of recessions (for example, those
associated with the labor market). Furthermore, not all the specific indicators have the same
behavior in each recession.
4.2 Select a set of highly coincident indicators
Although Figures 2 and 3 show that the set of indicators would be useful for ascertaining whether
a turning point has occurred, it is evident that there is considerable dispersion of the specific-
series turning points around the beginning and end of the recessions. As the multiple change-point
method assumes that the series being dated have been chosen to be coincident indicators, we
strongly recommend testing for this in advance by checking how synchronized the cycles are.
An automatic algorithm to date the reference cycle of the Spanish economy 11
For this purpose, we address the issue of synchronization between each indicator and the
SBCDC chronology by computing the index of concordance advocated by Harding and Pagan
(2002), which measures the percentage of time units spent in the same phase. The index, whose
values are plotted in Figure 4, confirms that there is a relatively high degree of synchronization
within the SBCDC business cycles. To ensure strong synchronization, we pick only those indicators
for which the concordance index exceeds 90%. Thus, the final set of indicators that exhibit the
highest correlation with the SBCDC chronology is formed by six time series: SSAI, SAI, ESIS,
ESIC, GDP and PC. This selection, which combines aggregated, sectorial, hard and soft indicators,
is not very different from the one commonly used by the NBER. Employment indicators are not
capable of pinpointing the Spanish cycle since they have a lagging behavior during expansions.
We use the SBCDC chronology to select the coincident indicators, given the high coincidence of
the modes detected in the set of indicators and the dating of the committee. Nevertheless, it should
be noted that our methodology does not require any prior information or the existence of any dating
committee. Alternatively, we could select the optimal set of coincident indicators with a search
algorithm applied to different combinations of indicators, where the optimal combination could be
selected on the basis of the number of clusters, the entropy and the variance of the confidence
intervals. This procedure could allow us to discard leading or lagging indicators and to choose the
combination of indicators that produces the largest modes. Since this option is computationally
more expensive, external information of experts, such as that of the SBCDC, can be used to speed
up the process.
It should be noted that, in any case, the choice of coinciding indicators, a key element for
the success of the procedure, requires the good judgment of the expert who must evaluate the
characteristics of each country, such as its productive structure, the quality and methodological
changes in the series or any other circumstance. Likewise, a qualitative assessment of the resulting
chronology would also be helpful. This is the reason to establish expert committees in different
countries.
Figure 5 provides a preliminary inspection of the individual chronologies of turning points by
plotting the kernel density estimator of the bivariate distribution of the resulting pairs of specific
peaks and troughs computed with the BBQ algorithm or the Markov-Switching procedure. The
distribution of turning points is multimodal, exhibiting various modes that cluster the individual
turning points around periods of SBCDC-referenced recoveries and declines. In particular, the
figure exhibits several modes around the seventies and the beginning of the eighties, the early
nineties, the first
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DE TRABAJO N.º 2139 of the new century and, finally, a large concentrated mode at the end
of the sample, in 2020. The figure also shows a larger variance of the turning point dates in the
seventies and early eighties, probably due to the lower quality of the indicators in the late years
of Franco’s dictatorship and the unstable political transition towards democracy. By contrast, thisYou can also read
